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Trigonometry plays a crucial role in understanding direction cosines. Direction cosines are the cosines of the angles that a line makes with the coordinate axes. These are noted as \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \) for angles with the \( x \), \( y \), and \( z \) axes, respectively.

When a line makes identical angles with two coordinate axes, the direction cosines with respect to those axes are the same. For example, in this exercise, \( \cos \alpha = \cos \gamma = \cos \theta \), where \( \theta \) is the angle with both the \( x \) and \( z \) axes. This symmetry greatly simplifies the problem.

Converting between trigonometric function values, like using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), is also essential for solving such problems. This concept helps relate different trigonometric expressions, making it easier to find the solution. So understanding these principles of trigonometry assists in tackling direction cosines effectively.

Understanding how angles relate to the coordinate axes is vital in directional analysis. Each axis represents a specific dimension in space – the \( x \), \( y \), and \( z \) axes. When a line forms an angle with any of these axes, it describes a particular orientation of the line in space.

In this exercise, the line forms a specific angle, \( \theta \), with the \( x \) and \( z \) axes, creating identical direction cosines for both axes, i.e., \( \cos \alpha \) and \( \cos \gamma \). The challenge arises when determining the angle with the \( y \)-axis, which impacts \( \cos \beta \), the direction cosine related to the \( y \)-axis.

The relationship between these angles and their cosines can be summed up in the equation \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \). This equation makes sure that the sum of the squares of the direction cosines of any line remains constant, giving a crucial point of reference in related problems. It ensures the spatial orientation is well-defined, and solutions are computable.

To solve complex mathematical problems, a structured approach is key. First, understanding the fundamental properties and relationships within the problem helps break it down systematically. For this exercise, knowing that the line has identical angles with two axes provides a starting point for such an analysis.

The solution process utilizes given conditions and mathematical identities to reach the final answer. For instance, here we used the relation \( \sin^2 \beta = 3 \sin^2 \theta \) in tandem with trigonometric identities to substitute and simplify into one cohesive equation. This helps us reach the required equation: \( 3 = 5\cos^2 \theta \).

After deriving the crucial equation, solving it becomes straightforward. Divide both sides of \( 3 = 5\cos^2 \theta \) by 5 to find \( \cos^2 \theta = \frac{3}{5} \). Checking this against possible answers confirms the solution's accuracy. Each step facilitates an organized path from understanding the context to deriving the solution, ensuring comprehension and capability in tackling similar problems.